Question

28. Derivation of Equation (7) in Example 4a. Show that the solution of the equationdi /dt + R/Li = V/Lisi = V/R + Cexp(-(R/L)i) .

Accepted Answer

Identify the given differential equation: $$\frac{di}{dt} + \frac{R}{L} i = \frac{V}{L}$$, which is a first-order linear ordinary differential equation in the variable $$i(t). $$Rewrite the equation in the standard linear form: $$\frac{di}{dt} + P(t) i = Q(t)$$, where $$P(t) = \frac{R}{L}$$ and $$Q(t) = \frac{V}{L}$$, both constants in this case. Find the integrating factor $$\mu(t)$$ using the formula $$\mu(t) = e$$^{\int P(t) dt}$$ = e$$^{\int \frac{R}$${L} dt} = e$$^{\frac{R}$${L} t}. $$Multiply both sides of the differential equation by the integrating factor to get: $$e^{\frac{R}$${L} t} \frac{di}{dt} + \frac{R}{L} $$e^{\frac{R}$${L} t} i = \frac{V}{L} $$e^{\frac{R}$${L} t}$$, which simplifies to $$\frac{d}{dt} \left( $$e^{\frac{R}$${L} t} i \right) = \frac{V}{L} $$e^{\frac{R}$${L} t}$$. Integrate both sides with respect to t$: $$\int \frac{d}{dt} \left( $$e^{\frac{R}$${L} t} i \right) dt = \int \frac{V}{L} $$e^{\frac{R}$${L} t} dt$$, then solve for i(t$$)$$ by dividing by e$$^{\frac{R}$${L} t}$$ and include the constant of integration $$C to $$obtain the general solution.

28. Derivation of Equation (7) in Example 4

a. Show that the solution of the equation
di /dt + R/Li = V/L
is
i = V/R + Cexp(-(R/L)i) .

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Concetti chiave

First-Order Linear Differential Equations

Integrating Factor Method

Solving for the Constant of Integration